Even indoors, plants can get bugs. There are things that you can use on them to help them get over infestations. My favorite is a spray of about a tablespoon of dish detergent and a quart of water and you spray those plants. And also a spray the soil surface. That cures most of my plant ills.
There is also Neem oil that you can spray.
I will tell you that spider plants are really really tough plants. I live in the semi-desert in California. I have bunches of them in hanging pots along my front porch and hundreds as ground covers on my hillsides. They can take some light frost without much damage. And lots of sun or lots of shade. When the babies break off they take root and I just weed
them out of where I'm not happy with them growing
. I do not always take the time to transplant elsewhere. When the potted ones begin to get rootbound I empty them out of the pots and lay them on their sides. I chop half the root ball off and shave off the sides a bit and plonk them back into the pots with new soil around. They have gone weeks sometimes without water and when they begin to droop I notice and turn water on them. They perk right back up again.
If you do cut off or lose babies, those are very easy to root into brand new plants. And the mamma plant will just send out new plantlets.